Applying Gauss' Law: Cylindrical Symmetry — Question 7

Question 7

An annular field profile determines bulk density and an unresolved core charge

In an infinite cylindrically symmetric system, the measured radial field is $E_r(r)=Ar+B/r$ only in the annulus $a<r<b$. A and B are constants. Nothing is specified about the charge profile in $r<a$.

1. Infer the volume density throughout the measured annulus.

2. Find the total charge per length in the unresolved core. Explain why B alone cannot automatically be identified with an actual line charge on the axis.

Question 7 - Solution

An annular field profile determines bulk density and an unresolved core charge

(a) Method and calculation

$$\lambda_{\mathrm{enc}}(r)=2\pi\epsilon_0rE_r=2\pi\epsilon_0(Ar^2+B).$$$$2\pi r\rho(r)=\frac{d\lambda_{\mathrm{enc}}}{dr}=4\pi\epsilon_0Ar\Rightarrow\rho=2\epsilon_0A.$$

(b) Interpretation and further calculation

$$\lambda_{\mathrm{core}}=\lambda_{\mathrm{enc}}(a^+)=2\pi\epsilon_0(Aa^2+B).$$$$\lambda_{\mathrm{enc}}(r)=\lambda_{\mathrm{core}}+\pi\rho(r^2-a^2).$$Only with extra information extending the same uniform bulk density toward the axis could $2\pi\epsilon_0B$ be assigned to an additional singular line source.

Physical check

A field measured on an annulus constrains enclosed totals but does not resolve the internal spatial arrangement of core charge.